Single particle energy diffusion from relativistic spontaneous localization
aa r X i v : . [ qu a n t - ph ] J un Single particle energy diffusion from relativistic spontaneouslocalization
D. J. Bedingham
Blackett LaboratoryImperial CollegeLondon SW7 2BZUK (Dated: September 12, 2018)
Abstract
Energy diffusion due to spontaneous localization (SL) for a relativistically-fast moving particleis examined. SL is an alternative to standard quantum theory in which quantum state reductionis treated as a random physical process which is incorporated into the Schr¨odinger equation inan observer-independent way. These models make predictions in conflict with standard quantumtheory one of which is non conservation of energy. On the basis of proposed relativistic extensionsof SL it is argued that for a single localized particle, non-relativistic SL should remain valid inthe rest frame of the particle. The implication is that relativistic calculations can be performedby transforming non-relativistic results from the particle rest frame to the frame of an inertialobserver. This is demonstrated by considering a relativistic stream of non-interacting particles ofcosmological origin and showing how their energy distribution evolves as a result of SL as theytraverse the Universe. A solution is presented and the potential for astrophysical observations isdiscussed. . INTRODUCTION Motivated by the measurement problem, spontaneous localization (SL) models are analternative to standard quantum theory in which quantum state reduction is treated as agenuine physical process [1, 2]. The typical formulation of these models is by modification ofthe Schr¨odinger equation to include non-linear and stochastic terms. Both of these featuresare well motivated. Stochasticity represents the random nature of state reduction. Nonlinearity enters since the probability of the state reducing to a particular outcome dependson the state itself (i.e. the Born rule). This approach is an empirical way of modellingthe behaviour of a quantum state as it is observed to behave in practice whether in ameasurement situation or not.The state-of-the-art non-relativistic SL model is the continuous spontaneous localization(CSL) model [3, 4]. This model is formulated in terms of a stochastic differential equationfor the state vector (see below) describing a continuous stochastic state trajectory in Hilbertspace. The CSL model reproduces the behaviour of non-relativistic quantum systems onthe micro scale whilst any macro superpositon of quasi-localized states is rapidly suppressed(the position basis takes a special role). This happens without having to make an arbitrarydivision between the micro and the macro domains - the theory itself determines when asuperpostion state is stable and when it is not. The remarkable thing is that this works ina way which is consistent with our experience.This paper concerns a side effect of SL models which is that they lead to a gradualincrease in the energy of a system on average. There are two complimentary ways in whichthis happens. The first is that as a result of the localization process, an initially spreadout wave function becomes narrower in position space and therefore broader in momentumspace. Since the free Hamiltonian is a convex function of momentum, the expected energyincreases. The second contribution is due to the fact that as the localizations happen thewave packet as a whole tends to undergo stochastic jumps in phase space (this is examined inmore detail below). This also results in spreading of momentum and consequently increasesthe expected energy on average.We shall consider this effect when the particle is travelling at relativistic speed withrespect to some observer. For the sake of definiteness we consider a stream of relativistically-fast non-interacting particles whose origin might be in the early universe. These particles2ravel at close to the speed of light and spend almost the whole lifetime of the Universetravelling freely until they eventually collide with an observer on Earth. The question wewish to ask is this: Would the non conservation of energy due to SL lead to a measurableeffect in the energy distribution of the observed particles?An obstacle in getting to this goal is that existing relativistic collapse models are com-plicated. In particular both the relativistic models of Refs[5] and [6] are non Markovianand this makes calculations more difficult to perform. We shall therefore take a short cut,arguing that the non-relativistic SL equations for a single particle should be valid in its restframe.A key feature of SL is that the localizations have an associated length scale. This is clearin discrete models such as the GRW model [7] where the state occasionally and randomlycollapses under the action of a Gaussian quasi projector in position space. The Gaussian hasan associated length scale defining its spread in space. This is essential. Were we to try toremove the length scale by making the localizations infinitely sharp the resulting collapsedwave function would have infinite energy. The same is true in CSL.To retain the feature of a fundamental length scale without resorting to the use of apreferred frame or foliation has proved to be one of the main difficulties when formulatingrelativistic extensions of SL. For example, in Ref.[5] the localizations are centred about arandom, time ordered sequence of points in spacetime, called flashes. The state at any givenstage is defined on an arbitrary spacelike surface to the future of the previous flash. The lawof the flashes requires that the state is unitarily evolved to a hyperbolic surface a random(Poisson distributed) timelike distance to the future of the previous flash; the state thendefines a probability distribution on the hyperbolic surface from which the location of anew flash is randomly drawn. The state is then modified by the action of a Gaussian quasiprojector with fixed length scale acting within the hyperbolic surface and centred on theflash location. We can deform the spacelike surface to the future of this new flash - the stateevolves unitarily until the next random flash happens. This procedure is independent of anyparticular frame of reference.Now consider two observers - one in the rest frame of the particle (which we take to bewell localized) and another moving with respect to it. For the rest frame observer, providedthat the proper time between flashes is sufficiently large compared to the length scale of thelocalization operator, the hyperboloid can be approximated by a plane and the model reduces3o the non-relativistic GRW model. However, due to the invariance of the construction, themoving observer will simply see the localization length contracted and the time betweenflashes dilated.A further example is provided by the model of Ref.[6]. This is a model involving quantumfields in which a new spacetime quantum field is introduced to mediate the influence of thelocalizations. The length scale arises from a smeared interaction between quantum fieldsand the mediating field. In order to avoid divergences, the smearing must be confined toa finite spacetime region. This is achieved by allowing the smearing function to dependon local properties of the quantum fields. The construction proposed results in a smearingfunction which defines a region in spacetime that is near to the point of interaction from thepoint of view of an averaged local rest frame.In each of these relativistic models the localizations can be said to happen with a fixedlength scale in the rest frame of the system (at least in the case of a well localized particlewhich we consider here where the rest frame is unambiguous). It seems natural on generalgrounds that this should be a universal feature of relativistic collapse models. A length scaleis necessary in order to define a localization, therefore a frame in which the length scaleapplies is necessary. The obvious way to do this without making reference to a preferredfoliation is to make use of local rest frames defined by the state. Whilst it might not beobvious precisely how to do this for a general state (perhaps requiring a fairly technicaldefinition as in both the relativistic models mentioned above), for a single particle in alocalized state the rest frame is clear.Relativistic invariance ensures that the localization process seen from a moving observer’spoint of view is simply a Lorentz transformation of events. We therefore take the approachof working in the particle rest frame where non-relativistic equations of motion are adequate,before transforming to the moving observers frame. Note that this is an assumption inspiredby relativistic considerations. An alternative possibility which we will not explore here is thatthere exists a preferred global frame (e.g. the cosmological frame) in which the localizationsoccur. In this case we might make the approximation that the non-relativistic CSL equationshold in only in this special frame.The structure of the paper is as follows: In Section II we show how the CSL model canbe significantly simplified in the case of a state describing a single localized particle. Wedescribe the steady state solution for the CSL model in this limit and reduce the behaviour4f the system to a classical diffusion in phase space. In Section III we derive the energydiffusion process and perform a transformation from the rest frame of the particle to theframe of an inertial observer moving at relativistic speed with respect to the particle. Wesolve this diffusion equation to find the probability distribution for the particle to end upwith a given energy. We conclude with some discussion in Section IV.
II. CSL IN THE LOCALIZED SINGLE PARTICLE LIMIT
In this section we organize the necessary non-relativistic results taken to hold in the restframe of the particle. We will demonstrate the steady state solution for the CSL modelin the case of a single, well localized particle. In order to do this we first demonstratethat the CSL equations can be simplified to a form known as QMUPL (quantum mechanicswith universal position localization) [8]. It is well known that relationships such as thisexist between the various SL models in certain limits [4, 8–10]. However, the followingdemonstration is believed to be novel.The CSL model concerns a quantum system described in terms of a non-relativisticquantum field. The state evolution is described by the stochastic differential equation d | ψ i = (cid:20) − i ~ ˆ H − λ Z d x (cid:16) ˆ N ( x ) − h ˆ N ( x ) i (cid:17) (cid:21) dt | ψ i + √ λ Z d x (cid:16) ˆ N ( x ) − h ˆ N ( x ) i (cid:17) dB t ( x ) | ψ i , (1)where the number density operator ˆ N ( x ) is given byˆ N ( x ) = (cid:16) απ (cid:17) / Z d x exp n − α x − y ) · ( x − y ) o ˆ a † ( y )ˆ a ( y ) , (2)the field annihilation and creation operators ˆ a ( x ) and ˆ a † ( x ) satisfy[ˆ a ( x ) , ˆ a † ( y )] = δ ( x − y ) , (3)and the field of Brownian motions satisfy E [ dB t ( x )] = 0; dB s ( x ) dB t ( y ) = δ st δ ( x − y ) dt. (4)The CSL parameters λ and 1 / √ α are understood respectively as the rate and the lengthscale of localization. The state remains normalized under Eq.(1) and we note that the5chr¨odinger equation is recovered in the limit that λ →
0. As mentioned in the Introductionthis model gives close agreement with the standard Schr¨odinger equation for micro systemsbut leads to rapid suppression of macro superpositions of quasi-localized states.Our task now is to simplify this construction for the purposes of our calculation. Wefirst assume that the state is describing a single particle. The approximation will be validalso if we a considering a swarm of non-interacting, non-overlapping, and non-entangledparticles where the state factorizes into single particle states. For a single particle the stateis represented by | ψ i = Z d x ψ ( x )ˆ a † ( x ) | i , (5)where | i is the vacuum state. Here we can identify ψ ( x ) as the wave function for the particle.Improper position eigenstates take the form | x i = ˆ a † ( x ) | i and the position operator is givenin terms of field creation and annihilation operators byˆ x = Z d x x ˆ a † ( x )ˆ a ( x ) . (6)Given this definition we find h ˆ x i = Z d x x | ψ ( x ) | , (7)as expected.Next we assume that the particle is sufficiently localized about a point ¯ y that we canmake the approximation (cf. Appendix B of [11])exp n − α x − y ) · ( x − y ) o ≃ exp n − α x − ¯ y ) · ( x − ¯ y ) o [1 + α ( x − ¯ y ) · ( y − ¯ y )] (8)This requires that | y − ¯ y | ≪ / √ α , i.e. the particle must be localized about the point ¯ y ona length scale much smaller than the localization length scale of the CSL model. The point¯ y is time dependent - it describes the location of the centre of the particle’s wave packet.6e can combine these various elements to calculate the following useful relations Z d x ˆ N ( x ) ≃ α y · ¯ y + α x · ˆ x − α ¯ y · ˆ x , Z d x h ˆ N ( x ) i ≃ α y · ¯ y + α h ˆ x i · h ˆ x i − α ¯ y · h ˆ x i , Z d x h ˆ N ( x ) i ˆ N ( x ) ≃ α y · ¯ y + α h ˆ x i · ˆ x − α y · h ˆ x i − α y · ˆ x , Z d x ˆ N ( x ) dB t ( x ) ≃ d ˜ B t + r α x − ¯ y ) · d B t , Z d x h ˆ N ( x ) i dB t ( x ) ≃ d ˜ B t + r α h ˆ x i − ¯ y ) · d B t , (9)where the Brownian motions B t and ˜ B t are related to the Brownian motion field through d B t = Z d x √ α (cid:16) απ (cid:17) / exp n − α x − ¯ y ) · ( x − ¯ y ) o ( x − ¯ y ) dB t ( x ) , (10) d ˜ B t = Z d x (cid:16) απ (cid:17) / exp n − α x − ¯ y ) · ( x − ¯ y ) o dB t ( x ) . (11)These definitions are readily shown to satisfy E [ dB i,t ] = 0 = E [ d ˜ B t ]; dB i,s dB j,t = δ ij δ st dt ; d ˜ B s d ˜ B t = δ st dt ; dB i,s d ˜ B t = 0 , (12)where dB i,t for i = 1 , , d B t relating to the or-thogonal spatial directions x i .Inserting the relations (9) into Eq.(1) results in d | ψ i = ( − i ~ ˆ Hdt − D ~ (ˆ x − h ˆ x i ) · (ˆ x − h ˆ x i ) dt + √ D ~ (ˆ x − h ˆ x i ) · d B t ) | ψ i , (13)with D given in terms of the CSL parameters as D = λα ~ , (14)(the factor of ~ is included for later convenience). In terms of momentum space fieldcreation and annihilation operators the Hamiltonian is given byˆ H = Z d p p · p m ˆ a † ( p )ˆ a ( p ) , (15)so that h p | ˆ H | ψ i = ( p · p / m ) ψ ( p ), where | p i = ˆ a † ( p ) | i , and ψ ( p ) = R dxe − i p · x / ~ ψ ( x ) / √ π ~ .We can therefore write ˆ H = ˆ p · ˆ p m , (16)7n the single particle case.Equation (13) is the QMULP model [8]. Not only is this equation useful as a scaled limitof CSL, it can also be used to describe the effect of a thermal environment when dealingwith open systems (see e.g. [12]).The advantage of Eq.(13) for our purpose is that we can calculate the steady state limitwhich occurs when the diffusive effects of the Hamiltonian balance with the localizing effectsof the SL terms [9, 13]. The form of the wave function in the steady state is given by ψ ( x ) = 1(2 πσ ∞ ) / exp (cid:26) − (1 − i )4 σ ∞ ( x − h ˆ x i ) · ( x − h ˆ x i ) + i ~ h ˆ p i · x (cid:27) , (17)where the steady state width is σ ∞ = r ~ Dm . (18)Starting from a general wave function it takes a time of order t loc ∼ p m ~ /D to reachthe steady state. The shape of the wave function is stable, however, the average positionand momentum of the packet undergoes 3D Brownian motion described by the followingdiffusion equations d h ˆ x i = h ˆ p i m dt + r ~ m d B t (19) d h ˆ p i = √ Dd B t . (20)These closed equations describe a classical Brownian motion for the wave packet.We now exhibit estimates of the steady state widths and localization times for a set ofdifferent species of particles. A detailed analysis of the valid range of choices for the SLparameters consistent with experiments can be found in Ref.[14]. Here we use the originalvalues suggested by GRW [7] of λ ∼ − s − and 1 / √ α ∼ − m . Based on experimentalevidence of spontaneous photon emission rates from Germanium it has been possible toargue that λ should increase with the mass of the particle as m [15]. This result we put inby hand (we would expect it to ultimately result from a more fundamental theory, perhapsinvolving gravity). We therefore assume that the GRW value relates to a single nucleon (asoriginally intended) and write λ = (cid:18) mm n (cid:19) × − s − , (21)where m n is a nucleon mass. Based on these assumptions the results are shown in thefollowing table: 8article σ ∞ t loc neutrino (0 . eV /c ) 1300 km yrs electron 12 m days proton 4 cm hrs Fe nucleus 2 mm hrs µm mins TABLE I: Steady state widths and localization times for various types of particle based onthe GRW parameters.We see from Table I that none of the particles satisfy the condition that σ ∞ ≪ / √ α for the GRW parameters. However, we note that many results relating to the effects ofspontaneous localization involve the parameters λ and α only in the combination λα (e.g.reduction rates, rate of energy increase). We therefore assume that the combination λα takes the GRW value (for a single nucleon) and that α is as large as necessary to fulfil thelocalized particle approximation made in this section.For a proton the value for σ ∞ = 4 cm gives a sense of being fairly well localized on largescales such that it can be treated like a particle, whilst at the same time very spread out onatomic scales such that its wavelike characteristics should dominate. III. RELATIVISTIC CSL
The localization times for the various particles in Table I indicate that on cosmologicaltime scales a free particle can be expected to exist in its steady state. We therefore assumethat the particle is in its steady state and concern ourselves only with the classical diffusivemotion of the packet. Since the behaviour is reduced to a classical form we write h ˆ x i = X and h ˆ p i = P . We assume that Eqs(19) and (20) apply in the rest frame of the particle -defined by P = 0 - whereby relativistic effects can be ignored. Note that due to the diffusivemotion of the particle the rest frame is continuously changing. Below we describe how totransform this process to the frame of an inertial observer.Denoting the rest frame O ′ and labelling coordinates in this frame with a prime we have9rom Eqs(19) and (20) d X ′ = r ~ m d B t ′ (22) d P ′ = √ Dd B t ′ . (23)We assume that the energy of the particle near its rest frame is given by the non-relativisticformula E ′ = P ′ · P ′ / m . Using Itˆo’s lemma and (12) we can derive the process for theenergy dE ′ = 3 Dm dt ′ + √ Dm P ′ · d B t ′ . (24)In the rest frame of the particle this is simply given by dE ′ = 3 Dm dt ′ . (25)The drift in energy is due to the stochastic shifts in momentum described by Eq.(23).We now transform from the particle rest frame to the cosmological frame O (coordinatesin this frame are unprimed) assumed to be an inertial frame in which the particle travels atrelativistic speed v ∼ c (we refer to this as the ultra-relativistic limit) in the X i direction.This direction will change as the particle undergoes diffusion - we rotate coordinates accord-ingly. The energy process in the cosmological frame is given by the Lorentz transformation dE = γdE ′ + vγdP ′ i ≃ γdE ′ + cγdP ′ i = 3 Dm γdt ′ + √ DcγdB i,t ′ , (26)with γ = 1 / p − v /c . This process is still expressed in terms of the rest frame time t ′ . Wewould like to express it in terms of the cosmological time t . To do this we use the Lorentztransformation dt = γ (cid:18) dt ′ − vdX ′ i c (cid:19) ≃ γ (cid:18) dt ′ − dX ′ i c (cid:19) . (27)Depending on the random fluctuations which define dX ′ i , dt can be positive or negative. Thisindicates that the particle in fact moves in tiny spacelike jumps. This makes the definition(27) unsuitable as the time parameter. A suitable definition can be guessed by consideringa finite period of cosmological time t = Z t ′ γds ′ − r ~ mc Z t ′ γdB i,s ′ . (28)10ote that γ is a function of t ′ (since the velocity v is a function of t ′ ). The two terms onthe right hand side can be thought of as a signal term and a noise term (with expectationzero). Then for a given value of t ′ we can estimate t by t = Z t ′ γds ′ . (29)The standard deviation in this estimate is given by the standard deviation of the Itˆo integralterm. This is given by σ t = ~ mc Z t ′ E [ γ ] ds ′ ! / . (30)So for γ approximately constant corresponding to a small amount of velocity dispersion wecan write E [ γ ] ∼ γ and the condition that σ t ≪ t corresponds to ~ mc ≪ t ′ . (31)For a proton ~ /mc ∼ − s so on cosmological time scales we can safely ignore the stochas-tic shifts of position of the wave packet in its rest frame.In order to convert to cosmological time we therefore write dt = γdt ′ and using a theoremof stochastic calculus we have dB t = γ / dB i,t ′ with dB t = dt (we drop the i since there isonly one Brownian motion factor in the energy process). Using E = γmc , Eq.(26) can nowbe written dE = 3 Dm dt + r Dm E dB t . (32)It is worth noting that this equation is remarkably similar to the non-relativistic equationthat is obtained from equation (24). We find that the only difference is an extra factor of √ dE = (cid:26) Dm − ˙ aa E (cid:27) dt + r Dm E dB t , (33)where a is the scale factor of the Universe and ˙ a/a , the Hubble parameter, is taken to bea constant. This equation describes the diffusion in energy of a particle due to CSL as ittravels at relativistic speed with respect to a cosmological observer (subject to the variousapproximations outlined earlier). 11q.(33) is in fact a Cox-Ingersoll-Ross (CIR) process [16]. The process is well known infinance where it is used to describe an instantaneous interest rate. A generic process of thistype dz = κ ( θ − z ) dt + σ √ zdB t , (34)has the properties that z is elastically pulled towards the long-term value θ at a rate de-termined by κ ; the origin is inaccessible if 2 κθ ≥ σ in which case z remains positive; thevariance in z increases as z increases; and there is a steady state distribution for z .How do these properties apply to the problem of relativistic energy diffusion? The condi-tion for the origin to be inaccessible is satisfied meaning that the energy will always remainpositive. We might expect that the energy would have a long term average value given by(3 D/m ) / ( ˙ a/a ) (independent of the initial state). However, the rate at which E is elasticallypulled to this value is ˙ a/a - of order of the inverse age of the Universe. Therefore for theenergy to reach its steady state distribution will require a time much longer than the age ofthe Universe (or more correctly, a much longer time than the time over which the Hubbleparameter can be approximated as a constant).The forward equation corresponding to Eg.(33) is given by ddt p t ( E | E ) = (cid:26) Dm E ∂ ∂E + (cid:18) ˙ aa − Dm (cid:19) ∂∂E + ˙ aa (cid:27) p t ( E | E ) . (35)We can compare this equation with a result from Refs.[17, 18] where a diffusion processin phase space due to a fundamental discreteness of spacetime is considered. There it isshown that starting from the idea of a random walk in momentum space, the condition ofLorentz invariance alone can be used to determine the form of the equation satisfied by theprobability distribution on phase space. It turns out that Eq.(35) is consistent with thisresult. In order to show this we find the equation satisfied by the marginal distribution formomentum and then convert from momentum to energy in the ultra-relativistic limit. Thisgives us confidence that our result (35) is at least relativistically correct. It also offers aninteresting point of comparison between SL models and spacetime discreteness.The advantage of working in the ultra-relativistic limit is that the forward equation (35)can be solved. With initial condition p ( E | E ) = δ ( E − E ) the probability distribution for E at time t conditional on a value E at time 0 is found to be [16] p t ( E | E ) = αβ EE e − α ( E − βE ) I (cid:16) α p βEE (cid:17) (36)12here α = ˙ aa mD − β , (37) β = exp (cid:26) − ˙ aa t (cid:27) , (38)and I is a modified Bessel function of the first kind of order 2. The expected value of E attime t is E t [ E | E ] = βE + 3 α , (39)and the variance is Var t [ E | E ] = 2 βα E + 3 α . (40)We can also write down the asymptotic limit of (36) valid when α √ βEE → ∞ p t ( E | E ) ∝ r m πDt (cid:18) E E (cid:19) / exp (cid:26) − mDt (cid:16) √ E − p E (cid:17) (cid:27) , (41)where we have assumed that t ≪ a/ ˙ a . And for completeness we give the steady statedistribution (despite the fact that this state is not achieved in the lifetime of the Universe) p ∞ ( E ) = 12 ω E e − ωE ′ , (42)where ω = ˙ aa mD . (43)The mean of the steady state distribution is 3 /ω and the variance is 3 /ω . Eq.(42) is infact the ultra-relativistic limit of the Maxwell-J¨uttner distribution describing the energiesof particles in an ideal gas in thermal equilibrium at relativistic temperatures.In Fig.1 we show the probability distribution for E given in Eq.(36). Once we fix unitssuch that E = 1 and make the assumption that t ≪ a/ ˙ a , the shape of the distributiondepends only on the value of the combination Dt/m . For large values ( > E ) the distributionis wide and flat whilst for small values ( < E ) the distribution becomes sharp and narrowtending towards a delta function at E = E as Dt/m → Dt/m . Wecan estimate
Dt/m using the GRW parameters along with (14) and (21). If we generously13 p t ( E | E ) Dt/m =0.01 ×E Dt/m =0.1 ×E Dt/m =1 ×E FIG. 1: Probability distribution of particle energy. See text for detailed description.assume that the particle has been freely travelling for almost the whole lifetime of theUniverse, t ∼ s , Dtm (cid:12)(cid:12)(cid:12)(cid:12)
GRW ∼ − mc . (44)We can make a more aggressive estimate by choosing λα ∼ m − s − - this is the orderof the current upper bound (CUB) on SL parameters imposed by diffraction experimentsusing large molecules [14]. Here using the same value for t we find Dtm (cid:12)(cid:12)(cid:12)(cid:12)
CUB ∼ − mc . (45)In both cases the value is very small. Given our assumption that the particle is travellingwith speed close to the speed of light its initial energy E must be at least several times therest energy. Since E ≫ Dt/m the distribution of energies will be highly peaked around E (see Fig.1) with variance of order DtE /m . Given the fact that it would be difficult to14dentify a source of massive free particles in the early Universe with very precise energy weconclude that it is very unlikely that this effect could be measured. IV. SUMMARY AND CONCLUSIONS
We have considered the case of a relativistically-fast moving particle with fixed initialenergy traversing the Universe over billions of years. Standard quantum theory predicts thatthe energy of the particle remains fixed and that the wave function slowly disperses in space.By contrast the CSL model, a modification of standard quantum theory to include quantumstate reduction as a dynamical process, predicts that the wave function remains localizedand that the energy of the particle undergoes diffusion. We have performed a calculation todetermine the distribution of possible kinetic energies obtained by the relativistic particleafter a long period of CSL evolution.Starting with the non-relativistic CSL model we have demonstrated that in the case of asufficiently-localized single particle the stochastic equations for the state vector can be recastin the simplified form of quantum mechanics with universal position localization (QMUPL).The properties of QMUPL are well understood. In particular there is a steady state form forthe wave packet achieved after a finite amount of free propagation. The average position andmomentum of the packet then satisfy a closed pair of coupled classical diffusion equations.This essentially reduces the complex quantum/stochastic behaviour of CSL to the simpleproblem of a classical diffusion in phase space.Although the non-relativistic CSL equations cannot be taken to apply for relativisticsystems we argued that they should be valid in the particle’s rest frame. This was based onexamining the form of proposed relativistic extensions of SL along with the general principlethat relativistic SL models should somehow incorporate a fixed localization length scale. Theobvious way to do this without the use of a preferred frame or foliation is to make referenceto local rest frames invariantly defined by the state. Roughly speaking this means that thelocalizations of a particle’s wave function will happen in the rest frame.We showed how the classical diffusion process satisfied by the steady state wave packetin the particle rest frame can be Lorantz transformed to describe the diffusion from thepoint of view of an inertial frame. This led us to derive a forward equation for the observedprobability distribution of energies in the case where the inertial observer sees the particle15ith speed v ∼ c . The energy process in this case is an example of a CIR process. Theseare well known from finance where they are useful for describing short rates. The forwardequation has a solution which we presented. In particular we were able to show that thesolution does not permit negative energies as we would expect.In fitting estimates for the CSL parameters to this energy distribution we found that evenusing the upper bound values obtained from diffraction experiments, the spread in energyand the average increase in energy due to the diffusion were both very small when comparedto the initial kinetic energy of the particle. Given that the initial energy of the particle inpractical examples will have some uncertainty, it is unlikely that the precision required tomeasure the relativistic energy diffusion due to CSL can be achieved. On the other handthis result means that the energy increases due to CSL are kept small, even on the scaleof the lifetime of the Universe and therefore do not pose a problem for the viability of thetheory.Given that the localizations which happen in the rest frame are Lorentz contracted fromthe perspective of a fast moving observer, one might have expected that the collapse effectswould be stronger for relativistic particles. However, this has to be balanced against thetime dilation effects which effectively reduce the localization rate from the observer’s pointof view. On the basis of the above calculation the two effects seem to cancel each other out.Relativistic particles do not obviously provide a way to amplify the effects of CSL for thepurpose of experimental test. ACKNOWLEDGEMENTS
I would like to thank Carlo Contaldi, Arttu Rajantie, Fay Dowker, and Philip Pearle forhelpful discussions and comments. [1] A. Bassi & GC. Ghirardi, Phys. Rept. 379 (2003).[2] A. Bassi et al , Rev. Mod. Phys. , 471 (2013).[3] P. Pearle, Phys. Rev. A , 2277 (1989).[4] GC. Ghirardi, P. Pearle, & A. Rimini, Phys. Rev. A , 78 (1990).[5] R. Tumulka, J. Stat. Phys. , 821 (2006)
6] D. J. Bedingham, Found. Phys.
686 (2011).[7] GC. Ghirardi, A. Rimini, & T. Weber, Phys. Rev. D , 470 (1986).[8] L. Di´osi, Phys. Rev. A , 1165 (1989).[9] P. Pearle, in: Quantum Theory: A Two-Time Success Story , Springer (2013); arXiv:1209.5082.[10] D. D¨urr, G. Hinrichs, & M. Kolb, J. Stat. Phys. , 1096 (2011).[11] B. Collett & P. Pearle, Found. Phys. , 1495 (2003).[12] J. J. Halliwell & A. Zoupas, Phys. Rev. D , 4697 (1997).[13] L. Di´osi, Phys. Lett. A , 233 (1988).[14] W. Feldmann & R. Tumulka, J. Phys. A: Math. Theor. , 06504 (2012).[15] P. Pearle & E. Squires, Phys. Rev. Lett. , 1 (1994).[16] J. Cox, J. Ingersoll, & S. Ross, Econometrica , 385 (1985).[17] F. Dowker, J. Henson, & R. Sorkin, Mod. Phys. Lett. A , 1829 (2004).[18] F. Dowker, L. Philpott, & R. Sorkin, Phys. Rev. D , 124047 (2009)., 124047 (2009).